BOREL αβ-MULTITRANSFORMS AND QUANTUM LERAYHIRSCH INTEGRAL REPRESENTATIONS OF SOLUTIONS OF QUANTUM DIFFERENTIAL EQUATIONS FOR P1-BUNDLES

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BOREL (α,β)-MULTITRANSFORMS AND QUANTUM LERAY–HIRSCH:

INTEGRAL REPRESENTATIONS OF SOLUTIONS OF QUANTUM

DIFFERENTIAL EQUATIONS FOR P1-BUNDLES

GIORDANO COTTI◦

◦Faculdade de Ciências da Universidade de Lisboa

Grupo de Física Matemática

Campo Grande Edifício C6, 1749-016 Lisboa, Portugal

Abstract. In this paper, we address the integration problem of the isomonodromic system

of quantum diﬀerential equations (qDEs) associated with the quantum cohomology of P1-

bundles on Fano varieties. It is shown that bases of solutions of the qDE of the total space of

the P1-bundle can be reconstructed from the datum of bases of solutions of the corresponding

qDE associated with the base space. This represents a quantum analog of the classical Leray–

Hirsch theorem in the context of the isomonodromic approach to quantum cohomology.

The reconstruction procedure of the solutions can be performed in terms of some integral

transforms, introduced in [Cot22], called Borel (α,β)-multitransforms. We emphasize the

emergence, in the explicit integral formulas, of an interesting sequence of special functions

(closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma

function) as integral kernels. Remarkably, these integral kernels have a universal feature,

being independent of the speciﬁcally chosen P1-bundle. When applied to projective bundles

on products of projective spaces, our results give Mellin–Barnes integral representations of

solutions of qDEs. As an example, we show how to integrate the qDE of blow-up of P2at

one point via Borel multitransforms of solutions of the qDE of P1.

Contents

1. Introduction 2

2. Gromov–Witten invariants and quantum cohomology 8

2.1. Notations 8

2.2. Gromov–Witten invariants 8

2.3. Mori and Kähler cones 9

2.4. Novikov ring, and Gromov–Witten potentials 10

2.5. Quantum cohomology as ΛX,ω-formal Frobenius manifold 11

2020 Mathematics Subject Classiﬁcation. 53D45, 44A20, 44A30, 33B20.

Key words and phrases. Quantum cohomology, projective bundles, Borel integral transform.

◦E-mail: gcotti@fc.ul.pt, gcotti@sissa.it

arXiv:2210.05445v2 [math.AG] 2 Oct 2024

2 GIORDANO COTTI

2.6. Quantum cohomology as C-formal Frobenius manifold 12

2.7. Quantum cohomology as Dubrovin–Frobenius manifold 13

3. QDEs, Borel multitransforms, Ekfunctions, and main results 14

3.1. Extended deformed connection 14

3.2. The quantum diﬀerential equation 15

3.3. Cyclic stratum, AΛ-stratum, master functions 15

3.4. Projective bundles: classical and quantum aspects 17

3.5. Analytic Borel multitransform 19

3.6. The functions Ek19

3.7. Main theorems 22

4. Proof of the main theorem 23

4.1. Topological–enumerative solution, and J-function 23

4.2. The Elezi–Brown theorem 24

4.3. The Ribenboim’s algebras Fκ(A)25

4.4. Formal Borel multitransform 27

4.5. The Ribenboim E-series of a projective bundle 28

4.6. Proof of Theorem 3.20 28

5. An example: blowing-up a point in P230

5.1. Classical and quantum cohomology 30

5.2. QDE and Λ-matrix 31

5.3. Solutions as Borel multitransforms 32

References 35

1. Introduction

1.1. Enumerative geometry is that branch of geometry that concerns the number of solutions

to a given geometrical problem, rather than explicitly ﬁnding them all. In the last decades,

ideas coming from physics brought innovation to enumerative geometry, with both new

techniques and the emergence of new rich geometrical structures. As an example, Gromov–

Witten theory, which focuses on counting numbers of curves on a target space, lead to the

discovery of quantum cohomology and the closely related quantum diﬀerential equations.

Given a smooth complex projective variety X, its quantum cohomology QH•(X)is a

family of commutative, associative, unital C-algebra structures on H•(X, C), obtained by

deforming the classical cohomological cup product. Such deformation is performed by adding

some “quantum correction terms”, containing information on the number of rational curves

on X. Namely, the structure constants of the quantum cohomology algebras can be ex-

pressed as third derivatives of a generating power series FX

0(t), with t= (t1, . . . , tn)and

n= dimCH•(X, C), of genus 0 Gromov–Witten invariants of X. Under the assumption of

convergence of FX

0(t)in some domain M⊆H•(X, C)∼

=Cn, the points t∈Mcan be used to

label the quantum algebra structures on H•(X, C), the corresponding product being denoted

by ◦t. This equips the quantum cohomology QH•(X)with an analytic Dubrovin–Frobenius

structure, with Mbeing the underlying complex manifold [Dub96,Man99,Her02,Sab08].

Points t∈Mare parameters of isomonodromic deformations of the quantum diﬀerential

equation (for short, qDE) of X. This is a system of linear diﬀerential equations of the form

dz ς(z, t) = U(t) + 1

zµ(t)ς(z, t),(1.1)

where ςis a z-dependent holomorphic vector ﬁeld1on M, and U,µare two endomorphisms of

the holomorphic tangent bundle of M. The ﬁrst operator Uis the operator of ◦-multiplication

by the Euler vector ﬁeld, a distinguished vector ﬁeld on M, obtained as perturbation of the

constant vector ﬁeld given by the ﬁrst Chern class c1(X). The second operator µ, called

grading operator, keeps track of the non-vanishing degrees of H•(X, C).

The qDE is a rich object associated with X. In the ﬁrst instance, the Gromov–Witten

theory of Xcan be reconstructed from the datum of the qDE (1.1) only. For details on

a Riemann–Hilbert–Birkhoﬀ approach to reconstruct the generating function FX

0(t), and

consequently the Dubrovin–Frobenius structure of QH•(X), see [Dub96,Dub99,Cot21a,

Cot21b]. In the second instance, the qDE of Xencodes not only information about the

enumerative (or symplectic) geometry of X, but also (conjecturally) about its topology and

complex geometry. In order to disclose such a great amount of information is via the study

of the asymptotics and the monodromy of its solutions, see [Dub98,GGI16,CDG18,Cot20].

The purpose of this paper is to construct new analytic tools, namely some integral repre-

sentations of the solutions of qDE, which will be particularly convenient to the study of

asymptotics, Stokes phenomena, and other analytical aspects. This will represent a contin-

uation of the research direction started in [Cot22].

1.2. The projectivization of vector bundles is one of the most natural constructions of smooth

projective varieties. Projective bundles have consequently been among the ﬁrst varieties

whose quantum cohomology algebras have been studied. See [QR98,CMR00,AM00].

The role played by the Gromov–Witten theory of projective bundles has recently been

revealed as central not only in the context of open deep conjectures (such as the crepant

transformation conjecture for ordinary ﬂops [LLW16a,LLW16b,LLQW16]) but even for

delicate foundational aspects of Gromov–Witten theory, such as its functoriality [LLW15].

The classical Leray–Hirsch theorem prescribes how to reconstruct the classical cohomology

algebra H•(P, C)of a projective bundle P=P(V)→Xon a variety X, from the knowledge

of the algebra H•(X, C)and the Chern roots ck(V),k= 0,...,rk V. These data only, indeed,

allow to write down an explicit presentation of H•(P, C). Several “quantum counterparts” of

this theorem have been proved over the years. Many of the main results proved in [MP06,

1Notice that tangent spaces of Mcan canonically be identiﬁed with H•(X, C).

4 GIORDANO COTTI

Ele05,Ele07,Bro14,LLW10,LLW15,LLW16a,LLW16b,Fan21] have a common thread:

they allow to deduce information about the quantum cohomology (or, more generally, the

Gromov–Witten theory) of a projective bundle P→Xstarting from information on the

quantum cohomology of X. See Section 3.4 for a more detailed discussion.

Following the same philosophy, in this paper we address the following question:

Q1. Is it possible to reconstruct a basis of solutions of the qDE of a projective bundle

P→Xfrom the datum of a basis of solutions of the qDE of X?

We obtain a positive result, under the assumption that Pis a Fano split P1-bundle on X.

In order to present the main results, we ﬁrst brieﬂy introduce some preliminary notions.

1.3. The ﬁrst notion we want to introduce is that of master function. We call master function

of Xat p∈QH•(X)any C-valued function Φς, holomorphic on the universal cover f

C∗of

C∗, of the form

Φς(z) = z−dimCX

2ZX

ς(z, p),

where ςis a solution of the qDE (1.1) specialized at t=p.

Rather than working directly with the space of solutions of the qDE, it is more convenient

to focus on the space Sp(X)of master functions of Xat p∈QH•(X). More precisely, in

addressing question Q1 above, we can split the problem into two parts:

Q2(1). Is it possible to reconstruct the space of solutions of the qDE (specialized at a point

p) from the datum of the space Sp(X)of master functions only?

Q2(2). Is it possible to reconstruct the space of master functions Sπ∗p(P)from the datum

of the space of master functions Sp(X), where p∈QH•(X)and 2π:P→X?

Question Q2(1) has been extensively studied in [Cot22]. In loc. cit., it is shown that the

answer is positive for generic p∈QH•(X). The problem of reducing a vector diﬀerential

equation to a scalar one is well known in the theory of ordinary diﬀerential equations. Such

a scalar reduction is equivalent to the choice of what is traditionally called a cyclic vector

for the diﬀerential system [Del70, Lemma II.1.3]. Moreover, several algorithmic reduction

procedures have been developed, see e.g. [Bar93] and references therein. On any Dubrovin–

Frobenius manifold M, we have a natural candidate for the cyclic vector, namely the unit

vector of the Frobenius algebras at each point p∈M. It turns out that such a choice is

working on the complement of an analytic subset of M, called AΛ-stratum [Cot22, Sec. 2].

More details will be given in Section 3.3. Consequently, question Q2(2) represents the main

problem to be still addressed.

The second notion we want to recall is that of (analytic)Borel (α,β)-multitransforms,

introduced in [Cot22]. These are C-multilinear integral transforms of tuples of analytic

functions. Given two h-tuples α,β∈(C∗)×h, with h⩾1, and an h-tuple of analytic

functions Φhon C\R<0, we deﬁne its Borel (α,β)-multitransform Bα,β[Φ1,...,Φh]by the

2The map πinduces a map π∗:H•(X, C)→H•(P, C). The class π∗p∈QH•(P)is the image of p.

integral

Bα,β[Φ1,...,Φh] (z) := 1

2π√−1ZH

j=1

Φjz

αjβjλ−βjeλdλ

λ,

where His a Hankel-type contour of integration, originating from −∞, circling the origin

once in the positive direction, and returning to −∞. See Figure 1.

Figure 1. Hankel-type contour of integration deﬁning Borel (α,β)-multitransform.

The third and last object we need to introduce for presenting our main results is a sequence

of special functions Ek, with k∈N. Consider the function E(s, z), analytic and single valued

on C×f

C∗, deﬁned by the integral

E(s, z) := zs

Γ(s)Z1

ts−1ez−ztdt, Re s > 0, z ∈f

C∗.

Alternatively, the function Ecan be deﬁned via the series expansion3

E(s, z) =

∞

k=0

zk+s

Γ(1 + k+s),(s, z)∈C×f

C∗.

We deﬁne the functions Ek∈ O(f

C∗),k∈N, as the iterated partial derivatives

Ek(z) := ∂k

∂skE(s, z)s=0

, k ⩾0.

For more explicit formulas for the Ek-functions, see Section 3.6.

The function Eis closely related to the upper and lower incomplete Gamma functions.

These are the functions Γ(s, z)and γ(s, z)deﬁned by the integrals

Γ(s, z) := Z∞

ts−1e−tdt, γ(s, z) := Zz

ts−1e−tdt, Re s > 0, z ∈C.

The incomplete Gamma functions were ﬁrst investigated, for real z, in 1811 by A.M. Legendre

[Leg11, Vol. 1, pp. 339–343 and later works]. The signiﬁcance of the decomposition Γ(s) =

γ(s, z) + Γ(s, z)was recognised by F.A. Prym in 1877 [Pry77], who seems to have been the

ﬁrst to investigate the functional behavior of these functions (that he denoted by Pand Q,

and which are often referred to as the Prym functions). The inconvenience of the function

γ(s, z)is not only of having poles at s= 0,−1,−2, . . . , but even of being multivalued in

3Hence, Ecan be thought as a “deformed” exponential function. This explains the notation E.

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BOREL(α,β)-MULTITRANSFORMSANDQUANTUMLERAY–HIRSCH:INTEGRALREPRESENTATIONSOFSOLUTIONSOFQUANTUMDIFFERENTIALEQUATIONSFORP1-BUNDLESGIORDANOCOTTI◦◦FaculdadedeCiênciasdaUniversidadedeLisboaGrupodeFísicaMatemáticaCampoGrandeEdifícioC6,1749-016Lisboa,PortugalAbstract.Inthispaper,weaddresstheintegrationproble...

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